from finite graphs to infinite; and beyondmaslar/talkgoet.pdf · 2013-03-01 · from finite graphs...
TRANSCRIPT
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
From finite graphs to infinite; and beyond
Agelos Georgakopoulos
Mathematisches SeminarUniversität Hamburg
Göttingen, 5.3.2009
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theorems
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theorems
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Hamilton cycles
Hamilton cycle: A cycle containing all vertices.
Some examples:
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theorems
Extremal graph theorymany others ...
⇒ need more general notions
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theoremsExtremal graph theory
many others ...
⇒ need more general notions
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theoremsExtremal graph theorymany others ...
⇒ need more general notions
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theoremsExtremal graph theorymany others ...
⇒ need more general notions
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Spanning Double-Rays
Classical approach to ‘save’ Hamilton cycle theorems:accept double-rays as infinite cycles
......
This approach only extends finite theorems in very restricted cases:
Theorem (Tutte ’56)Every finite 4-connected planar graph has aHamilton cycle
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Spanning Double-Rays
Classical approach to ‘save’ Hamilton cycle theorems:accept double-rays as infinite cycles
......
This approach only extends finite theorems in very restricted cases:
Theorem (Tutte ’56)Every finite 4-connected planar graph has aHamilton cycle
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Spanning Double-Rays
Classical approach to ‘save’ Hamilton cycle theorems:accept double-rays as infinite cycles
......
This approach only extends finite theorems in very restricted cases:
Theorem (Tutte ’56)Every finite 4-connected planar graph has aHamilton cycle
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Spanning Double-Rays
Classical approach: accept double-rays as infinite cycles
......
This approach only extends finite theorems in very restricted cases:
Theorem (Yu ’05)Every locally finite 4-connected planargraph has a spanning double ray ...
unlessit is 3-divisible.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Spanning Double-Rays
Classical approach: accept double-rays as infinite cycles
......
This approach only extends finite theorems in very restricted cases:
Theorem (Yu ’05)Every locally finite 4-connected planargraph has a spanning double ray ... unlessit is 3-divisible.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Compactifying by Points at Infinity
A 3-divisible graph
can have no spanning double ray
......
... but a Hamilton cycle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Compactifying by Points at Infinity
A 3-divisible graphcan have no spanning double ray
......
... but a Hamilton cycle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Compactifying by Points at Infinity
A 3-divisible graphcan have no spanning double ray
......
... but a Hamilton cycle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Compactifying by Points at Infinity
A 3-divisible graphcan have no spanning double ray
......
... but a Hamilton cycle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Ends
end: equivalence class of raystwo rays are equivalent if no finite vertex set separates them
......
two ends
one end
... ... ... ...... ... ... ...
uncountably many ends
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Ends
end: equivalence class of raystwo rays are equivalent if no finite vertex set separates them
......
two ends
one end
... ... ... ...... ... ... ...
uncountably many ends
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Ends
end: equivalence class of raystwo rays are equivalent if no finite vertex set separates them
......
two ends
one end
... ... ... ...... ... ... ...
uncountably many ends
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Ends
end: equivalence class of raystwo rays are equivalent if no finite vertex set separates them
......
two ends
one end
... ... ... ...... ... ... ...
uncountably many ends
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The End Compactification
|G|
Every ray converges to its end
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The End Compactification
|G|
Every ray converges to its end
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The End Compactification
|G|
Every ray converges to its end
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The End Compactification
|G|
Every ray converges to its end
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
(Equivalent) definition of |G|
Give each edge e a length `(e)
This naturally induces a metric d` on G
Denote by |G|` the completion of (G, d`)
Theorem (G ’06)
If∑
e∈E(G) `(e) < ∞ then |G|` is homeomorphic to |G|.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
(Equivalent) definition of |G|
Give each edge e a length `(e)
This naturally induces a metric d` on G
Denote by |G|` the completion of (G, d`)
Theorem (G ’06)
If∑
e∈E(G) `(e) < ∞ then |G|` is homeomorphic to |G|.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
(Equivalent) definition of |G|
Give each edge e a length `(e)
This naturally induces a metric d` on G
Denote by |G|` the completion of (G, d`)
Theorem (G ’06)
If∑
e∈E(G) `(e) < ∞ then |G|` is homeomorphic to |G|.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
(Equivalent) definition of |G|
Give each edge e a length `(e)
This naturally induces a metric d` on G
Denote by |G|` the completion of (G, d`)
Theorem (G ’06)
If∑
e∈E(G) `(e) < ∞ then |G|` is homeomorphic to |G|.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
Hamilton circle:a circle containing all vertices,
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
Hamilton circle:a circle containing all vertices
,
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
Hamilton circle:a circle containing all vertices
,
(and all ends?)
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
Hamilton circle:a circle containing all vertices, and thus also all ends.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
Hamilton circle:a circle containing all vertices, and thus also all ends.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
the wild circle of Diestel & Kühn
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Infinite Cycles
Circle:A homeomorphic image of S1 in |G|.
the wild circle of Diestel & Kühn
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Fleischner’s Theorem
Theorem (Fleischner ’74)The square of a finite 2-connected graph has aHamilton cycle
Theorem (Thomassen ’78)The square of a locally finite 2-connected 1-endedgraph has a Hamilton circle.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Fleischner’s Theorem
Theorem (Fleischner ’74)The square of a finite 2-connected graph has aHamilton cycle
Theorem (Thomassen ’78)The square of a locally finite 2-connected 1-endedgraph has a Hamilton circle.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The Theorem
Theorem (G ’06)The square of any locally finite 2-connectedgraph has a Hamilton circle
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Proof?
Hilbert’s space filling curve:
a sequence of injective curves with a non-injective limit
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The Theorem
Theorem (G ’06)The square of any locally finite 2-connectedgraph has a Hamilton circle
CorollaryCayley graphs are “morally” hamiltonian.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The Theorem
Theorem (G ’06)The square of any locally finite 2-connectedgraph has a Hamilton circle
CorollaryCayley graphs are “morally” hamiltonian.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Hamiltonicity in Cayley graphs
Problem (Rapaport-Strasser ’59)Does every finite connected Cayley graph have a Hamiltoncycle?
ProblemDoes every connected 1-ended Cayley graph have a Hamiltoncircle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Hamiltonicity in Cayley graphs
Problem (Rapaport-Strasser ’59)Does every finite connected Cayley graph have a Hamiltoncycle?
ProblemDoes every connected 1-ended Cayley graph have a Hamiltoncircle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Hamiltonicity in Cayley graphs
Problem (Rapaport-Strasser ’59)Does every finite connected Cayley graph have a Hamiltoncycle?
ProblemDoes every connected 1-ended Cayley graph have a Hamiltoncircle?
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Hamiltonicity in Cayley graphs
Problem (Rapaport-Strasser ’59)Does every finite connected Cayley graph have a Hamiltoncycle?
ProblemDoes every connected 1-ended Cayley graph have a Hamiltoncircle?
ProblemProve that every connected Cayley graph of a finitely generatedgroup Γ has a Hamilton circle unless Γ is the amalgamatedproduct of more than k groups over a subgroup of order k.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Things that go wrong in infinite graphs
Many finite theorems fail for infinite graphs:
Hamilton cycle theoremsExtremal graph theorymany others ...
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Extremal Graph Theory
Theorem (Mader ’72)Any finite graph of minimum degree at least4k has a k-connected subgraph.
k -connected means: you can delete any k − 1 vertices and the graphwill still be connected.
Theorem (M. Stein ’05)
Let k ∈ N and let G be a locally finite graph such that every vertexhas degree at least 6k2 − 5k + 3 and every end has degree at least6k2 − 9k + 4. Then G has a k-connected subgraph.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Extremal Graph Theory
Theorem (Mader ’72)Any finite graph of minimum degree at least4k has a k-connected subgraph.
k -connected means: you can delete any k − 1 vertices and the graphwill still be connected.
Theorem (M. Stein ’05)
Let k ∈ N and let G be a locally finite graph such that every vertexhas degree at least 6k2 − 5k + 3 and every end has degree at least6k2 − 9k + 4. Then G has a k-connected subgraph.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Cycle Space
The cycle space C(G) of a finite graph:A vector space over Z2
Consists of all sums of cycles
i.e., the first simplicial homology group of G.
The topological cycle space C(G) of a locally finite graph G isdefined similarly but:
Allows edge sets of infinite circles;Allows infinite sums (whenever well-defined).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Cycle Space
The cycle space C(G) of a finite graph:A vector space over Z2
Consists of all sums of cyclesi.e., the first simplicial homology group of G.
The topological cycle space C(G) of a locally finite graph G isdefined similarly but:
Allows edge sets of infinite circles;Allows infinite sums (whenever well-defined).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Cycle Space
The cycle space C(G) of a finite graph:A vector space over Z2
Consists of all sums of cyclesi.e., the first simplicial homology group of G.
The topological cycle space C(G) of a locally finite graph G isdefined similarly but:
Allows edge sets of infinite circles;Allows infinite sums (whenever well-defined).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Cycle Space
The cycle space C(G) of a finite graph:A vector space over Z2
Consists of all sums of cyclesi.e., the first simplicial homology group of G.
The topological cycle space C(G) of a locally finite graph G isdefined similarly but:
Allows edge sets of infinite circles;
Allows infinite sums (whenever well-defined).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Cycle Space
The cycle space C(G) of a finite graph:A vector space over Z2
Consists of all sums of cyclesi.e., the first simplicial homology group of G.
The topological cycle space C(G) of a locally finite graph G isdefined similarly but:
Allows edge sets of infinite circles;Allows infinite sums (whenever well-defined).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
The topological Cycle Space
Known facts:A connected graph has an Euler tour iffevery edge-cut is even (Euler)G is planar iff C(G) has a simplegenerating set (MacLane)If G is 3-connected then its peripheralcycles generate C(G) (Tutte)
Generalisations:
Bruhn & Stein
Bruhn & Stein
Bruhn
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
MacLane’s Planarity Criterion
Theorem (MacLane ’37)
A finite graph G is planar iff C(G) has asimple generating set.
simple: no edge appears in more than two generators.
Theorem (Bruhn & Stein’05)... verbatim generalisation for locally finite G
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
MacLane’s Planarity Criterion
Theorem (MacLane ’37)
A finite graph G is planar iff C(G) has asimple generating set.
simple: no edge appears in more than two generators.
Theorem (Bruhn & Stein’05)... verbatim generalisation for locally finite G
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
C(G) and Singular Homology
There is a canonical homomorphism
f : H1(|G|) → C(G)
Theorem (Diestel & Sprüssel ’07)f is surjective but not injective.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
C(G) and Singular Homology
There is a canonical homomorphism
f : H1(|G|) → C(G)
Theorem (Diestel & Sprüssel ’07)f is surjective but not injective.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
C(G) and Singular Homology
There is a canonical homomorphism
f : H1(|G|) → C(G)
Theorem (Diestel & Sprüssel ’07)f is surjective but not injective.
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
C(G) and Singular Homology
ProblemModify H1 to obtain a homology theory thatcaptures C(G) when applied to |G| and generalisesgraph-theoretical theorems to arbitrary continua.
In particular:
ProblemCharacterise the continua embeddable in theplane
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
C(G) and Singular Homology
ProblemModify H1 to obtain a homology theory thatcaptures C(G) when applied to |G| and generalisesgraph-theoretical theorems to arbitrary continua.
In particular:
ProblemCharacterise the continua embeddable in theplane
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Electrical Networks
An electrical network is a graph G with an assignment of resistancesr : E(G) → R+, and two special vertices (source – sink) pumping aflow of constant value I into the network.
electrical flow: A flow satisfying Kirchhoff’s second law (for finitecycles.
If G is finite then the electrical flow is unique, if it is infinite thenthere might be several; but:
Theorem (G ’08)
If∑
e∈E r(e) < ∞ then there is a uniquenon-elusive electrical flow of finite energy.
energy :=∑
e∈E i2(e)r(e).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Electrical Networks
An electrical network is a graph G with an assignment of resistancesr : E(G) → R+, and two special vertices (source – sink) pumping aflow of constant value I into the network.
electrical flow: A flow satisfying Kirchhoff’s second law (for finitecycles.
If G is finite then the electrical flow is unique, if it is infinite thenthere might be several; but:
Theorem (G ’08)
If∑
e∈E r(e) < ∞ then there is a uniquenon-elusive electrical flow of finite energy.
energy :=∑
e∈E i2(e)r(e).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Electrical Networks
An electrical network is a graph G with an assignment of resistancesr : E(G) → R+, and two special vertices (source – sink) pumping aflow of constant value I into the network.
electrical flow: A flow satisfying Kirchhoff’s second law (for finitecycles.
If G is finite then the electrical flow is unique, if it is infinite thenthere might be several; but:
Theorem (G ’08)
If∑
e∈E r(e) < ∞ then there is a uniquenon-elusive electrical flow of finite energy.
energy :=∑
e∈E i2(e)r(e).
Agelos Georgakopoulos Infinite graphs
Introduction Topological cycles Fleischner’s Theorem Extremal Graph Theory The Cycle Space Electrical Networks
Electrical Networks
An electrical network is a graph G with an assignment of resistancesr : E(G) → R+, and two special vertices (source – sink) pumping aflow of constant value I into the network.
electrical flow: A flow satisfying Kirchhoff’s second law (for finitecycles.
If G is finite then the electrical flow is unique, if it is infinite thenthere might be several; but:
Theorem (G ’08)
If∑
e∈E r(e) < ∞ then there is a uniquenon-elusive electrical flow of finite energy.
energy :=∑
e∈E i2(e)r(e).
Agelos Georgakopoulos Infinite graphs